Secure Probabilistic Analytics Using Homomorphic Encryption

ABSTRACT

An example method for performing a secure probabilistic analytic includes acquiring, by a client, an analytic, at least one analytic parameter associated with the analytic, and an encryption scheme. The encryption scheme can include a public key for encryption and a private key for decryption. The method further includes generating, using the encryption scheme, at least one analytical vector based on the analytic and analytic parameter, and sending the analytical vector and the encryption scheme to at least one server. The method includes generating, by the server based on the encryption scheme, a set of terms from a data set, evaluating the analytical vector over the set of terms to obtain an encrypted result; estimating, by the server, a probabilistic error of the encrypted result; and sending, by the server, the encrypted result and the probabilistic error to the client where the encrypted result is decrypted.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit and priority of U.S. ProvisionalApplication Ser. No. 62/448,890, filed on Jan. 20, 2017; U.S.Provisional Application Ser. No. 62/448,918, filed on Jan. 20, 2017;U.S. Provisional Application Ser. No. 62/448,893, filed on Jan. 20,2017; U.S. Provisional Application Ser. No. 62/448,906, filed on Jan.20, 2017; U.S. Provisional Application Ser. No. 62/448,908, filed onJan. 20, 2017; U.S. Provisional Application Ser. No. 62/448,913, filedon Jan. 20, 2017; U.S. Provisional Application Ser. No. 62/448,916,filed on Jan. 20, 2017; U.S. Provisional Application Ser. No.62/448,883, filed on Jan. 20, 2017; U.S. Provisional Application62/448,885, filed on Jan. 20, 2017; U.S. Provisional Application Ser.No. 62/448,902, filed on Jan. 20, 2017; U.S. Provisional ApplicationSer. No. 62/448,896, filed on Jan. 20, 2017; U.S. ProvisionalApplication Ser. No. 62/448,899, filed on Jan. 20, 2017; and U.S.Provisional Application Ser. No. 62/462,818, filed on Feb. 23, 2017, allof which are hereby incorporated by reference herein, including allreferences and appendices, for all purposes.

TECHNICAL FIELD

This disclosure relates to the technical field of encryption anddecryption of data. More specifically, this disclosure relates tosystems and methods for performing secure probabilistic analytics usinga homomorphic encryption.

BACKGROUND

With development of computer technologies, many sensitive data, such asfinancial information and medical records can be kept on remote serversor cloud-based computing resources. Authorized users can access thesensitive data using applications running, for example, on theirpersonal computing devices. Typically, personal computing devices areconnected, via data networks, to servers or cloud-based computingresources. Therefore, the sensitive data can be subject to unauthorizedaccess.

Encryption techniques, such as a homomorphic encryption, can be appliedto the sensitive data to prevent unauthorized access. The encryptiontechniques can be used to protect “data in use”, “data in rest”, and“data in transit”. A homomorphic encryption is a form of encryption inwhich a specific algebraic operation (generally referred to as additionor multiplication) performed on plaintext, is equivalent to anotheroperation performed on ciphertext. For example, in Partially HomomorphicEncryption (PHE) schemes, multiplication in ciphertext is equal toaddition of the same values in plaintext.

SUMMARY

This summary is provided to introduce a selection of concepts in asimplified form that are further described in the Detailed Descriptionbelow. This summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used as an aid in determining the scope of the claimed subjectmatter.

Generally, the present disclosure is directed to the technology forsecure data processing. Some embodiments of the present disclosure mayfacilitate a secure transmission of analytics from a client device toremote computing resource(s) for performing analytics over a data sourceand a secure transmission of results of the analytics from the computingresources back to the client device.

According to one example embodiment of the present disclosure, a methodfor performing a secure probabilistic analysis using homomorphicencryption is provided. The method may include receiving, from a client,by at least one server, at least one analytic vector, a term generationfunction, and a keyed hash function. The at least one analytic vectorcan be encrypted using a homomorphic encryption scheme. The homomorphicencryption scheme can include a public key for encryption and a privatekey for decryption. The method may further include extracting, by the atleast one server, a set of term components from a data set using theterm generation function and the keyed hashed function. The method mayfurther include evaluating, by the at least one server, the at least oneanalytic vector over the set of term components to obtain at least oneencrypted result. The method may further include estimating, by the atleast one server, a probabilistic error of the at least one encryptedresult. The method may allow sending, by the at least one server, the atleast one encrypted result and the probabilistic error to the client,wherein the client is configured to decrypt the at least one encryptedresult using the homomorphic encryption scheme.

In some embodiments, the homomorphic encryption scheme includes apartially homomorphic encryption scheme. The partially homomorphicencryption scheme may include at least one of the following: a Rivest,Shamir and Adleman cryptosystem, Elgamal cryptosystem, Benalohcryptosystem, Goldwasser-Micali cryptosystem, and Pallier cryptosystem.In certain embodiments, the homomorphic encryption scheme may include afully homomorphic encryption scheme.

In some embodiments, the at least one analytic vector is generated basedon an analytic and at least one parameter associated with the analytic.The generation of the analytic vector may include extracting, using theterm generation function, a set of term elements from the analytic andthe at least one analytic parameter. The generation of the analyticvector may further include generating, using the keyed hash function,the set of hashes from the set of term elements. The generation mayfurther include determining whether an index of at least one element ofthe analytical vector is present in the set of hashes. If the index ispresent in the set of hashes, the at least one element is assigned anon-zero value. The non-zero value can include an encrypted value of anon-zero bitmask of a term element selected from the set of termelements, wherein the hash of the term element is equal to the index.The encrypted value can be obtained using the homomorphic encryptionscheme. If the index is not present in the set of the hashes, the atleast one element is assigned a zero value.

In certain embodiments, a dimension of the at least one analytic vectoris greater than the number of elements in the set of the term elements.In some embodiments, estimating the probabilistic error is performedbased on a hash collision rate of the hash function over the data setand a dimension of the at least one analytic vector. In variousembodiments, the data set can be in a plaintext form, deterministicallyencrypted, and/or semantically encrypted.

According to one example embodiment of the present disclosure, a systemfor performing a secure probabilistic analysis using homomorphicencryption is provided. The system may include at least one processorand a memory storing processor-executable codes, wherein the at leastone processor can be configured to implement the operations of theabove-mentioned method for performing a secure probabilistic analysisusing homomorphic encryption.

According to yet another example embodiment of the present disclosure,the operations of the above-mentioned method for performing secureprobabilistic analytics using a homomorphic encryption are stored on amachine-readable medium comprising instructions, which when implementedby one or more processors perform the recited operations.

Other example embodiments of the disclosure and aspects will becomeapparent from the following description taken in conjunction with thefollowing drawings.

BRIEF DESCRIPTION OF DRAWINGS

Exemplary embodiments are illustrated by way of example and notlimitation in the figures of the accompanying drawings, in which likereferences indicate similar elements.

FIG. 1 is a block diagram of an example environment suitable forpracticing methods for secure probabilistic analytics using ahomomorphic encryption as described herein.

FIG. 2 is a block diagram showing details of a homomorphic encryptionscheme, according to an example embodiment.

FIG. 3 is a flow chart of an example method for performing secureprobabilistic analytics using a homomorphic encryption.

FIG. 4 is a computer system that can be used to implement someembodiments of the present disclosure.

DETAILED DESCRIPTION OF EXAMPLARY EMBODIMENTS

The technology disclosed herein is concerned with methods and systemsfor performing secure probabilistic analytics over data source using ahomomorphic encryption. Embodiments of the present disclosure mayfacilitate a secure transmission of analytics from a client device tocomputing resource(s) providing a target data source and securetransmission of results of analytics from the computing resource(s) backto the client device.

Some embodiments of the present disclosure may be used to encrypt ananalytic on a client device using homomorphic encryption techniques. Theencrypted analytic can be sent to computing resource(s) providingdesired data source(s). The encrypted analytics can be performed overdesired data source(s) to produce encrypted results. The encryptedresults can be returned to the client device and decrypted using thehomomorphic encryption techniques. Embodiments of the present disclosuremay allow performing of an analytic over desired data sources in asecure and private manner because neither content of the analytic norresults of the analytic are revealed to a data owner, observer, orattacker.

According to one example embodiment of the present disclosure, a methodfor performing secure probabilistic analytics using a homomorphicencryption may commence with acquiring, by a client, an analytic, atleast one analytic parameter associated with the analytic, and anencryption scheme. The encryption scheme may include a public key forencryption and a private key for decryption. The method may furtherinclude generating, by the client and using the encryption scheme, atleast one analytical vector based on the analytic and the at least oneanalytic parameter. The method may further include sending, by theclient, the at least one analytical vector and the encryption scheme, toat least one server.

The method may also include acquiring, by the at least one server, adata set for performing the analytic. The method may allow extracting,by the at least one server and based on the encryption scheme, a set ofterms from the data set. The method may further include, evaluating, bythe at least one server, the at least one analytical vector over the setof terms to obtain at least one encrypted result. The method may furtherallow estimating, by the at least one server, a probabilistic error ofthe at least one encrypted result. The method may also include sending,by the at least one server, the at least one encrypted result and theprobabilistic error to the client. The method may also includedecrypting, by the client and based on the encryption scheme, the atleast one encrypted result to generate at least one result of theanalytic.

Referring now to the drawings, various embodiments are described inwhich like reference numerals represent like parts and assembliesthroughout the several views. It should be noted that the reference tovarious embodiments does not limit the scope of the claims attachedhereto. Additionally, any examples outlined in this specification arenot intended to be limiting and merely set forth some of the manypossible embodiments for the appended claims.

FIG. 1 shows a block diagram of an example environment 100 suitable forpracticing the methods described herein. It should be noted, however,that the environment 100 is just one example and is a simplifiedembodiment provided for illustrative purposes, and reasonable deviationsof this embodiment are possible as will be evident for those skilled inthe art.

As shown in FIG. 1, the environment 100 may include at least one clientdevice 105 (also referred to as a client 105) and at least one server110. The client(s) 105 can include any appropriate computing devicehaving network functionalities allowing the device to communicate toserver(s) 110. In some embodiments, the client(s) 105 can be connectedto the server(s) 110 via one or more wired or wireless communicationsnetworks. In various embodiments, the client 105 includes, but is notlimited to, a computer (e.g., laptop computer, tablet computer, desktopcomputer), a server, cellular phone, smart phone, gaming console,multimedia system, smart television device, set-top box, infotainmentsystem, in-vehicle computing device, informational kiosk, smart homecomputer, software application, computer operating system, modem,router, and so forth. In some embodiments, the client(s) 105 can be usedby users for Internet browsing purposes.

In some embodiments, the server(s) 110 may be configured to store orprovide access to at least one data source 115. In certain embodiments,the server(s) 110 may include a standalone computing device. In variousembodiments, the data source(s) 115 may be located on a single server110 or distributed over multiple server(s) 110. The data source(s) 115may include plaintext data, deterministically encrypted data,semantically encrypted data, or a combination of thereof.

In some embodiments, the server(s) 110 may be implemented as cloud-basedcomputing resource shared by multiple users. The cloud-based computingresource(s) can include hardware and software available at a remotelocation and accessible over a network (for example, the Internet). Thecloud-based computing resource(s) can be dynamically re-allocated basedon demand. The cloud-based computing resources may include one or moreserver farms/clusters including a collection of computer servers whichcan be co-located with network switches and/or routers.

In various embodiments, the client(s) 105 can make certain clientinquires within the environment 100. For example, the client(s) 105 maybe configured to send analytics to the server(s) 110 to be performedover the data source(s) 115. The server(s) 110 can be configured toperform the analytics over the data source(s) 115 and return the resultsof analytics to the client(s) 105.

To protect the content of the analytics, the client(s) 105 can beconfigured to encrypt the analytics using a homomorphic encryptionscheme. The homomorphic encryption scheme can include a partiallyhomomorphic encryption scheme and a fully homomorphic encryption scheme.The partially homomorphic encryption scheme can include one of a Rivest,Shamir and Adleman cryptosystem, Elgamal cryptosystem, Benalohcryptosystem, Goldwasser-Micali cryptosystem, and Pallier cryptosystem.The analytics can be encrypted with a public (encryption) key of thehomomorphic encryption scheme. The encrypted analytics and the publickey can be sent to the server 110. The encrypted analytics can be onlydecrypted with a private (decryption) key of the homomorphic encryptionscheme. The decryption key can be kept on the client 105 and neverprovided to the server(s) 110.

To protect the content of the results of the analytic, the server(s) 110can be configured to perform the encrypted analytics on the data sourceusing the same homographic encryption scheme and the public key receivedfrom the client 105 and, thereby, obtain encrypted results of theanalytics. The encrypted results can be sent to the client(s) 105. Theclient(s) 105 can decrypt the encrypted results using the private key.Because the private key is always kept on the client(s) 105, neitherencrypted analytic nor encrypted results of the analytics can bedecrypted on the server 110 or when intercepted while in transitionbetween the client(s) 105 and the server(s) 110.

FIG. 2 is a block diagram 200 showing details of homomorphic encryptionscheme 200, according to some example embodiments. The modules of thescheme 200 can be implemented as software instructions stored in memoryof the client 105 and executed by at least one processor of the client105. The client 105 may be configured to acquire a desired analytic A tobe executed over data source 115. The analytic A can be associated withanalytic parameter set {A_P}. The analytic A and analytic parameter set{A_P} can be further encrypted into a sequence of homomorphic analyticalvectors {A_V} using a homomorphic encryption scheme E.

The scheme 200 may include a term generation (TG) function 210. The termgeneration function 210 can be used to extract a set of term elements{T} of analytic A that correspond to an analytic parameter A_P. For,example, if the analytic parameter A_P is a frequency distribution fordatabase elements in <row:column> pairs where row=Y, then the set {T}reflects the frequency distribution of these elements from the database.

The scheme 200 may further include a keyed hash function H(T) 220. Thehash function H(T) can be used to obtain a set H(T)={H(T): T in {T}}.The set H(T) is the range of the hash function H(T) over the set of termelements {T}. The keyed hash function H(T) can be associated with apublic key used for the encryption. The number of distinct elements inthe set H(T) is equal to the number of distinct elements in the set ofterm elements {T}.

The scheme 200 may further include an analytical vector constructionmodule 230. The module 230 can be used to construct an analytical vectorA_V for the analytic parameter A_P. The desired size s of the analyticalvector A_V can be selected to be greater than the number of distinctelements in the set of term elements {T}. For index j=0, . . . , (s−1):if H(T)=j for a term element T in the set {T}, then vector componentA_V[j]=E(B_j) where B_j is a nonzero bit mask corresponding to the termelement T, wherein E is the homographic encryption scheme. If there isno T in {T} such that H(T)=j, then A_V[j]=E(0). In this manner, theanalytical vector A_V includes encryptions of nonzero bitmasks for onlythe term elements present in the set {T}. The analytic A cannot berecovered from the analytical vectors {A_V} without a private keyassociated with the homomorphic encryption scheme E.

The client 105 can be further configured to send the analytical vectors{A_V}, the term generation function TG, and the hash function H(T) withthe public key to the server(s) 110.

In some embodiments, the server(s) 110 can be configured to extract aset of term elements {T} from the data source 115 using the termgeneration function TG and the keyed hash function H(T). The server(s)110 can be further configured to evaluate the encrypted analyticalvectors {A_V} over the set of term elements {T} to produce encryptedresults E(R). The server(s) 110 can be further configured to estimate aprobabilistic error b of the encrypted results E(R) based on a hashcollision rate of the hash function H(T) over data source 115 anddimension of analytical vectors {A_V}. The server(s) 110 can be furtherconfigured to send the encrypted results E(R) and the probabilisticerror b to the client 105.

The client 105 can be configured to decrypt the encrypted results E(R)in order to obtain the results R using the private key of thehomomorphic encryption scheme E. Because the analytical vector {A_V}includes nonzero entries for terms in set {T}, the homomorphicproperties of E ensure that only results corresponding to the nonzeroelements of the analytical vector {A_V} are present in results R.

FIG. 3 is a flow chart of an example method 300 for performing secureprobabilistic analytics using a homomorphic encryption, according tosome example embodiments. The method 300 may be performed withinenvironment 100 illustrated in FIG. 1. Notably, the steps recited belowmay be implemented in an order different than described and shown in theFIG. 3. Moreover, the method 300 may have additional steps not shownherein, but which can be evident to those skilled in the art from thepresent disclosure. The method 300 may also have fewer steps thanoutlined below and shown in FIG. 3.

The method 300 may commence in block 305 with receiving, by at least oneserver, from a client, at least one analytic vector, a term generationfunction, and a keyed hash function. The at least one analytic vectorcan be encrypted using the homomorphic encryption scheme. Thehomomorphic encryption scheme may include a public key for encryptionand a private key for decryption.

In block 310, the method 300 may proceed with extracting, by the atleast one server, a set of term components from a data set using theterm generation function and the keyed hashed function.

In block 315, the method 300 may evaluate, by the at least one server,the at least one analytic vector over the set of term components toobtain at least one encrypted result.

In block 320, the method 300 may estimate, by the at least one server, aprobabilistic error of the at least one encrypted result. The estimatecan be based on a hash collision of the keyed hash function over thedata set and the length of the analytic vector.

In block 325, the method may proceed with sending, by the at least oneserver, the at least one encrypted result and the probabilistic error tothe client. The client can be configured to decrypt the at least oneencrypted result using the homomorphic encryption scheme.

FIG. 4 illustrates an exemplary computer system 400 that may be used toimplement some embodiments of the present disclosure. The computersystem 400 of FIG. 4 may be implemented in the contexts of the likes ofthe client 105, the server(s) 110, and the data source 115. The computersystem 400 of FIG. 4 includes one or more processor units 410 and mainmemory 420. Main memory 420 stores, in part, instructions and data forexecution by processor units 410. Main memory 420 stores the executablecode when in operation, in this example. The computer system 400 of FIG.4 further includes a mass data storage 430, portable storage device 440,output devices 450, user input devices 460, a graphics display system470, and peripheral devices 480.

The components shown in FIG. 4 are depicted as being connected via asingle bus 490. The components may be connected through one or more datatransport means. Processor unit 410 and main memory 420 is connected viaa local microprocessor bus, and the mass data storage 430, peripheraldevice(s) 480, portable storage device 440, and graphics display system470 are connected via one or more input/output (I/O) buses.

Mass data storage 430, which can be implemented with a magnetic diskdrive, solid state drive, or an optical disk drive, is a non-volatilestorage device for storing data and instructions for use by processorunit 410. Mass data storage 430 stores the system software forimplementing embodiments of the present disclosure for purposes ofloading that software into main memory 420.

Portable storage device 440 operates in conjunction with a portablenon-volatile storage medium, such as a flash drive, floppy disk, compactdisk, digital video disc, or Universal Serial Bus (USB) storage device,to input and output data and code to and from the computer system 400 ofFIG. 4. The system software for implementing embodiments of the presentdisclosure is stored on such a portable medium and input to the computersystem 400 via the portable storage device 440.

User input devices 460 can provide a portion of a user interface. Userinput devices 460 may include one or more microphones, an alphanumerickeypad, such as a keyboard, for inputting alphanumeric and otherinformation, or a pointing device, such as a mouse, a trackball, stylus,or cursor direction keys. User input devices 460 can also include atouchscreen. Additionally, the computer system 400 as shown in FIG. 4includes output devices 450. Suitable output devices 450 includespeakers, printers, network interfaces, and monitors.

Graphics display system 470 include a liquid crystal display (LCD) orother suitable display device. Graphics display system 470 isconfigurable to receive textual and graphical information and processesthe information for output to the display device.

Peripheral devices 480 may include any type of computer support deviceto add additional functionality to the computer system.

The components provided in the computer system 400 of FIG. 4 are thosetypically found in computer systems that may be suitable for use withembodiments of the present disclosure and are intended to represent abroad category of such computer components that are well known in theart. Thus, the computer system 400 of FIG. 4 can be a personal computer(PC), hand held computer system, telephone, mobile computer system,workstation, tablet, phablet, mobile phone, server, minicomputer,mainframe computer, wearable, or any other computer system. The computermay also include different bus configurations, networked platforms,multi-processor platforms, and the like. Various operating systems maybe used including UNIX, LINUX, WINDOWS, MAC OS, PALM OS, QNX ANDROID,IOS, CHROME, TIZEN, and other suitable operating systems.

The processing for various embodiments may be implemented in softwarethat is cloud-based. In some embodiments, the computer system 400 isimplemented as a cloud-based computing environment, such as a virtualmachine operating within a computing cloud. In other embodiments, thecomputer system 400 may itself include a cloud-based computingenvironment, where the functionalities of the computer system 400 areexecuted in a distributed fashion. Thus, the computer system 400, whenconfigured as a computing cloud, may include pluralities of computingdevices in various forms, as will be described in greater detail below.

In general, a cloud-based computing environment is a resource thattypically combines the computational power of a large grouping ofprocessors (such as within web servers) and/or that combines the storagecapacity of a large grouping of computer memories or storage devices.Systems that provide cloud-based resources may be utilized exclusivelyby their owners or such systems may be accessible to outside users whodeploy applications within the computing infrastructure to obtain thebenefit of large computational or storage resources.

The cloud may be formed, for example, by a network of web servers thatcomprise a plurality of computing devices, such as the computer system400, with each server (or at least a plurality thereof) providingprocessor and/or storage resources. These servers may manage workloadsprovided by multiple users (e.g., cloud resource customers or otherusers). Typically, each user places workload demands upon the cloud thatvary in real-time, sometimes dramatically. The nature and extent ofthese variations typically depends on the type of business associatedwith the user.

The present technology is described above with reference to exampleembodiments. Therefore, other variations upon the example embodimentsare intended to be covered by the present disclosure.

What is claimed is:
 1. A method for performing a secure probabilisticanalysis using homomorphic encryption, the method comprising: receiving,from a client, by at least one server, at least one analytic vector, theat least one analytic vector being encrypted using a homomorphicencryption scheme, a term generation function, and a keyed hashfunction, wherein the homomorphic encryption scheme includes a publickey for encryption and a private key for decryption; extracting, by theat least one server, a set of term components from a data set using theterm generation function and the keyed hashed function; evaluating, bythe at least one server, the at least one analytic vector over the setof term components to obtain at least one encrypted result; estimating,by the at least one server, a probabilistic error of the at least oneencrypted result; and sending, by the at least one server, the at leastone encrypted result and the probabilistic error to the client, whereinthe client is configured to decrypt the at least one encrypted resultusing the homomorphic encryption scheme.
 2. The method of claim 1,wherein the homomorphic encryption scheme includes a partiallyhomomorphic encryption scheme.
 3. The method of claim 2, wherein thepartially homomorphic encryption scheme includes at least one of thefollowing: a Rivest, a Shamir and Adleman cryptosystem, an Elgamalcryptosystem, a Benaloh cryptosystem, a Goldwasser-Micali cryptosystem,and a Pallier, cryptosystem.
 4. The method of claim 1, wherein thehomomorphic encryption scheme includes a fully homomorphic encryptionscheme.
 5. The method of claim 1, wherein the at least one analyticvector is generated based on an analytic and at least one parameterassociated with the analytic, the generation of the at least oneanalytic vector including: extracting, using the term generationfunction, a set of term elements from the analytic and the at least oneanalytic parameter; generating, using the keyed hash function, a set ofhashes from the set of term elements; determining whether an index of atleast one element of the analytical vector is present in the set ofhashes; if the index is present in the set of hashes, assigning the atleast one element a non-zero value; and if the index is not present inthe set of hashes, assigning the at least one element a zero value. 6.The method of claim 5, wherein a dimension of the at least one analyticvector is greater than the number of elements in the set of termelements.
 7. The method of claim 5, wherein the non-zero value is anencrypted value of a non-zero bitmask of a term element of the set ofterm elements, the hash of the term element being equal to the index,the encrypted value being obtained using the homomorphic encryptionscheme.
 8. The method of claim 1, wherein estimating the probabilisticerror is based on a hash collision rate of the keyed hash function overthe data set and dimension of the at least one analytic vector.
 9. Themethod of claim 1, wherein the data set is in a plaintext form.
 10. Themethod of claim 1, wherein the data set is deterministically encryptedor semantically encrypted.
 11. A system for performing a secureprobabilistic analysis using homomorphic encryption, the systemcomprising: at least one processor; and a memory communicatively coupledwith the at least one processor, the memory storing instructions, whichwhen executed by the at least processor perform a method comprising:receiving, from a client, at least one analytic vector, the at least oneanalytic vector being encrypted using a homomorphic encryption scheme, aterm generation function, and a keyed hash function, wherein thehomomorphic encryption scheme includes a public key for encryption and aprivate key for decryption; extracting, by the at least one server, aset of term components from a data set using the term generationfunction and the keyed hashed function; evaluating the at least oneanalytic vector over the set of term components to obtain at least oneencrypted result; estimating a probabilistic error of the at least oneencrypted result; and sending the at least one encrypted result and theprobabilistic error to the client, wherein the client is configured todecrypt the at least one encrypted result using the homomorphicencryption scheme.
 12. The system of claim 11, wherein the homomorphicencryption scheme includes a partially homomorphic encryption scheme.13. The system of claim 11, wherein the homomorphic encryption schemeincludes at least one of a Rivest, a Shamir and Adleman cryptosystem, anElgamal cryptosystem, a Benaloh cryptosystem, a Goldwasser-Micalicryptosystem, and a Pallier cryptosystem
 14. The system of claim 11,wherein the homomorphic encryption scheme includes a fully homomorphicencryption scheme.
 15. The system of claim 11, wherein the at least oneanalytic vector is generated based on an analytic and at least oneparameter associated with the analytic and the generation includes:extracting, using the term generation function, the set of term elementsfrom the analytic and the at least one analytic parameter; generating,using the keyed hash function, a set of hashes from the set of termelements; determining whether an index of at least one element of theanalytical vector is present in the set of hashes; if the index ispresent in the set of hashes, assigning the at least one element anon-zero value; and if the index is not present in the set of hashes,assigning the at least one element a zero value.
 16. The system of claim15, wherein a dimension of the at least one analytic vector is greaterthan the number of elements in the set of term elements.
 17. The systemof claim 15, wherein the non-zero value is an encrypted value of anon-zero bitmask of a term element of the set of term elements, the hashof the term element being equal to the index, the encrypted value beingobtained using the homomorphic encryption scheme.
 18. The system ofclaim 11, wherein estimating the probabilistic error is based on a hashcollision rate of the keyed hash function over the data set anddimension of the at least one analytic vector.
 19. The system of claim11, wherein the data set is in one of the following forms: a plaintext,deterministically encrypted, and semantically encrypted.
 20. Anon-transitory computer-readable storage medium having embodied thereoninstructions, which when executed by at least one processor, performsteps of a method, the method comprising: receiving, by at least oneserver from a client, at least one analytic vector, the at least oneanalytic vector being encrypted using a homomorphic encryption scheme, aterm generation function, and a keyed hash function, wherein thehomomorphic encryption scheme includes a public key for encryption and aprivate key for decryption; extracting, by the at least one server, aset of term components from a data set using the term generationfunction and the keyed hashed function; evaluating, by the at least oneserver, the at least one analytic vector over the set of term componentsto obtain at least one encrypted result; estimating, by the at least oneserver, a probabilistic error of the at least one encrypted result; andsending, by the at least one server, the at least one encrypted resultand the probabilistic error to the client, wherein the client isconfigured to decrypt the at least one encrypted result using thehomomorphic encryption scheme.